Welcome to our lesson on ratio and proportion. A ratio is a comparison of two quantities of the same unit. It shows how much of one quantity there is compared to another. For example, if we have 3 apples and 2 oranges, the ratio of apples to oranges is 3 to 2, written as 3:2. A proportion is an equation that states two ratios are equal.
There are three common ways to express ratios. First, using a colon, such as 3:2. Second, as a fraction, written as 3 over 2. Third, in words, saying "3 to 2". All three forms represent exactly the same relationship between the two quantities. For our example of 3 apples to 2 oranges, we can write this as 3:2, 3/2, or "3 to 2".
Now let's define proportion. A proportion is an equation that states two ratios are equal. If a over b equals c over d, then a, b, c, and d are said to be in proportion. This means the relationship between a and b is exactly the same as the relationship between c and d. For example, 2 over 3 equals 4 over 6. We can verify this by cross multiplication: 2 times 6 equals 12, and 3 times 4 also equals 12. Since both sides equal 12, this confirms our proportion is correct.
Let's work through a practical example. In a recipe, the ratio of flour to sugar is 4 to 3. If we use 8 cups of flour, how much sugar do we need? First, we set up a proportion: 4 over 3 equals 8 over x, where x is the unknown amount of sugar. Next, we cross multiply: 4 times x equals 3 times 8, which gives us 4x equals 24. Finally, we solve for x by dividing both sides by 4: x equals 24 divided by 4, which equals 6. Therefore, we need 6 cups of sugar.
Let's summarize what we've learned about ratio and proportion. A ratio is a comparison of two quantities that can be written as a colon, fraction, or in words. A proportion states that two ratios are equal, and we can solve proportions using cross multiplication. These concepts are widely used in everyday life, including cooking recipes, reading maps, creating scale models, and many other real-world applications. Understanding ratios and proportions is fundamental for solving problems involving relationships between quantities.